# Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:

${\displaystyle \delta _{ij}={\begin{cases}0&{\text{if }}i\neq j,\\1&{\text{if }}i=j.\end{cases}}}$

This latter case is for convenience.

## Properties

The following equations are satisfied:

{\displaystyle {\begin{aligned}\sum _{j}\delta _{ij}a_{j}&=a_{i},\\\sum _{i}a_{i}\delta _{ij}&=a_{j},\\\sum _{k}\delta _{ik}\delta _{kj}&=\delta _{ij}.\end{aligned}}}

Therefore, the matrix δ can be considered as an identity matrix.

Another useful representation is the following form:

${\displaystyle \delta _{nm}={\frac {1}{N}}\sum _{k=1}^{N}e^{2\pi i{\frac {k}{N}}(n-m)}}$

This can be derived using the formula for the finite geometric series.

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